%------------------------------------------------------------------------------
% File : Twee---2.7
% Problem : REL026+2 : TPTP v9.3.1. Released v4.0.0.
% Transfm : none
% Format : tptp:raw
% Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% Computer : n001.cluster.edu
% Model : x86_64 x86_64
% CPU : Intel(R) Xeon(R) CPU E5-2620 v4 2.10GHz
% Memory : 8046.5625MB
% OS : Linux 6.8.0-71-generic
% CPULimit : 300s
% WCLimit : 300s
% DateTime : Tue Sep 29 12:33:15 PM UTC 2026
% Result : Theorem 19.41s 2.92s
% Output : Proof 20.74s
% Verified :
% SZS Type : -
% Comments :
%------------------------------------------------------------------------------
%----WARNING: Could not form TPTP format derivation
%------------------------------------------------------------------------------
%----ORIGINAL SYSTEM OUTPUT
% 0.00/0.03 % Problem : REL026+2 : TPTP v9.3.1. Released v4.0.0.
% 0.00/0.04 % Command : run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 0.08/0.35 % Computer : n001.cluster.edu
% 0.08/0.35 % Model : x86_64 x86_64
% 0.08/0.35 % CPU : Intel(R) Xeon(R) CPU E5-2620 v4 @ 2.10GHz
% 0.08/0.35 % Memory : 8046.5625MB
% 0.08/0.35 % OS : Linux 6.8.0-71-generic
% 0.08/0.35 % CPULimit : 300
% 0.08/0.35 % WCLimit : 300
% 0.08/0.35 % DateTime : Sun Sep 27 22:58:46 UTC 2026
% 0.08/0.36 % CPUTime :
% 0.08/0.36 Running run_twee /export/starexec/sandbox2/benchmark/theBenchmark.p
% 19.41/2.92 Command-line arguments: --no-flatten-goal
% 19.41/2.92
% 19.41/2.92 % SZS status Theorem
% 19.41/2.92
% 19.41/3.02 % SZS output start Proof
% 19.41/3.02 Axiom 1 (def_zero): zero = meet(X, complement(X)).
% 19.41/3.02 Axiom 2 (converse_idempotence): converse(converse(X)) = X.
% 19.41/3.02 Axiom 3 (composition_identity): composition(X, one) = X.
% 19.41/3.02 Axiom 4 (maddux1_join_commutativity): join(X, Y) = join(Y, X).
% 19.41/3.02 Axiom 5 (def_top): top = join(X, complement(X)).
% 19.41/3.02 Axiom 6 (goals): join(x0, one) = one.
% 19.41/3.02 Axiom 7 (maddux4_definiton_of_meet): meet(X, Y) = complement(join(complement(X), complement(Y))).
% 19.41/3.02 Axiom 8 (converse_multiplicativity): converse(composition(X, Y)) = composition(converse(Y), converse(X)).
% 19.41/3.02 Axiom 9 (composition_associativity): composition(X, composition(Y, Z)) = composition(composition(X, Y), Z).
% 19.41/3.02 Axiom 10 (converse_additivity): converse(join(X, Y)) = join(converse(X), converse(Y)).
% 19.41/3.02 Axiom 11 (maddux2_join_associativity): join(X, join(Y, Z)) = join(join(X, Y), Z).
% 19.41/3.02 Axiom 12 (composition_distributivity): composition(join(X, Y), Z) = join(composition(X, Z), composition(Y, Z)).
% 19.41/3.02 Axiom 13 (maddux3_a_kind_of_de_Morgan): X = join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y))).
% 19.41/3.02 Axiom 14 (converse_cancellativity): join(composition(converse(X), complement(composition(X, Y))), complement(Y)) = complement(Y).
% 19.41/3.02
% 19.41/3.02 Lemma 15: complement(top) = zero.
% 19.41/3.02 Proof:
% 19.41/3.02 complement(top)
% 19.41/3.02 = { by axiom 5 (def_top) }
% 19.41/3.02 complement(join(complement(X), complement(complement(X))))
% 19.41/3.02 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 19.41/3.02 meet(X, complement(X))
% 19.41/3.02 = { by axiom 1 (def_zero) R->L }
% 19.41/3.02 zero
% 19.41/3.02
% 19.41/3.02 Lemma 16: join(meet(X, Y), complement(join(complement(X), Y))) = X.
% 19.41/3.02 Proof:
% 19.41/3.02 join(meet(X, Y), complement(join(complement(X), Y)))
% 19.41/3.02 = { by axiom 7 (maddux4_definiton_of_meet) }
% 19.41/3.02 join(complement(join(complement(X), complement(Y))), complement(join(complement(X), Y)))
% 19.41/3.02 = { by axiom 13 (maddux3_a_kind_of_de_Morgan) R->L }
% 19.41/3.03 X
% 19.41/3.03
% 19.41/3.03 Lemma 17: join(meet(X, Y), meet(X, complement(Y))) = X.
% 19.41/3.03 Proof:
% 19.41/3.03 join(meet(X, Y), meet(X, complement(Y)))
% 19.41/3.03 = { by axiom 4 (maddux1_join_commutativity) R->L }
% 19.41/3.03 join(meet(X, complement(Y)), meet(X, Y))
% 19.41/3.03 = { by axiom 7 (maddux4_definiton_of_meet) }
% 19.41/3.03 join(meet(X, complement(Y)), complement(join(complement(X), complement(Y))))
% 19.41/3.03 = { by lemma 16 }
% 19.41/3.03 X
% 19.41/3.03
% 19.41/3.03 Lemma 18: meet(Y, X) = meet(X, Y).
% 19.41/3.03 Proof:
% 19.41/3.03 meet(Y, X)
% 19.41/3.03 = { by axiom 7 (maddux4_definiton_of_meet) }
% 19.41/3.03 complement(join(complement(Y), complement(X)))
% 19.41/3.03 = { by axiom 4 (maddux1_join_commutativity) R->L }
% 19.41/3.03 complement(join(complement(X), complement(Y)))
% 19.41/3.03 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 19.41/3.03 meet(X, Y)
% 19.41/3.03
% 19.41/3.03 Lemma 19: complement(join(zero, complement(X))) = meet(X, top).
% 19.41/3.03 Proof:
% 19.41/3.03 complement(join(zero, complement(X)))
% 19.41/3.03 = { by lemma 15 R->L }
% 19.41/3.03 complement(join(complement(top), complement(X)))
% 19.41/3.03 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 19.41/3.03 meet(top, X)
% 19.41/3.03 = { by lemma 18 R->L }
% 19.41/3.03 meet(X, top)
% 19.41/3.03
% 19.41/3.03 Lemma 20: composition(converse(one), X) = X.
% 19.41/3.03 Proof:
% 19.41/3.03 composition(converse(one), X)
% 19.41/3.03 = { by axiom 2 (converse_idempotence) R->L }
% 19.41/3.03 composition(converse(one), converse(converse(X)))
% 19.41/3.03 = { by axiom 8 (converse_multiplicativity) R->L }
% 19.41/3.03 converse(composition(converse(X), one))
% 19.41/3.03 = { by axiom 3 (composition_identity) }
% 19.41/3.03 converse(converse(X))
% 19.41/3.03 = { by axiom 2 (converse_idempotence) }
% 19.41/3.03 X
% 19.41/3.03
% 19.41/3.03 Lemma 21: composition(one, X) = X.
% 19.41/3.03 Proof:
% 19.41/3.03 composition(one, X)
% 19.41/3.03 = { by lemma 20 R->L }
% 19.41/3.03 composition(converse(one), composition(one, X))
% 19.41/3.03 = { by axiom 9 (composition_associativity) }
% 19.41/3.03 composition(composition(converse(one), one), X)
% 19.41/3.03 = { by axiom 3 (composition_identity) }
% 19.41/3.03 composition(converse(one), X)
% 19.41/3.03 = { by lemma 20 }
% 19.41/3.03 X
% 19.41/3.03
% 19.41/3.03 Lemma 22: join(complement(X), complement(X)) = complement(X).
% 19.41/3.03 Proof:
% 19.41/3.03 join(complement(X), complement(X))
% 19.41/3.03 = { by lemma 20 R->L }
% 19.41/3.03 join(complement(X), composition(converse(one), complement(X)))
% 19.41/3.03 = { by lemma 21 R->L }
% 19.41/3.03 join(complement(X), composition(converse(one), complement(composition(one, X))))
% 19.41/3.03 = { by axiom 4 (maddux1_join_commutativity) R->L }
% 19.41/3.03 join(composition(converse(one), complement(composition(one, X))), complement(X))
% 19.41/3.03 = { by axiom 14 (converse_cancellativity) }
% 19.41/3.03 complement(X)
% 19.41/3.03
% 19.41/3.03 Lemma 23: join(zero, complement(complement(X))) = X.
% 19.41/3.03 Proof:
% 19.41/3.03 join(zero, complement(complement(X)))
% 19.41/3.03 = { by axiom 1 (def_zero) }
% 19.41/3.03 join(meet(X, complement(X)), complement(complement(X)))
% 19.41/3.03 = { by lemma 22 R->L }
% 19.41/3.03 join(meet(X, complement(X)), complement(join(complement(X), complement(X))))
% 19.41/3.03 = { by lemma 16 }
% 19.41/3.03 X
% 19.41/3.03
% 19.41/3.03 Lemma 24: meet(top, complement(X)) = complement(X).
% 19.41/3.03 Proof:
% 19.41/3.03 meet(top, complement(X))
% 19.41/3.03 = { by lemma 18 }
% 19.41/3.03 meet(complement(X), top)
% 19.41/3.03 = { by lemma 19 R->L }
% 19.41/3.03 complement(join(zero, complement(complement(X))))
% 19.41/3.03 = { by lemma 23 }
% 19.41/3.04 complement(X)
% 19.41/3.04
% 19.41/3.04 Lemma 25: complement(zero) = top.
% 19.41/3.04 Proof:
% 19.41/3.04 complement(zero)
% 19.41/3.04 = { by lemma 17 R->L }
% 19.41/3.04 join(meet(complement(zero), top), meet(complement(zero), complement(top)))
% 19.41/3.04 = { by lemma 15 }
% 19.41/3.04 join(meet(complement(zero), top), meet(complement(zero), zero))
% 19.41/3.04 = { by axiom 4 (maddux1_join_commutativity) }
% 19.41/3.04 join(meet(complement(zero), zero), meet(complement(zero), top))
% 19.41/3.04 = { by lemma 18 R->L }
% 19.41/3.04 join(meet(zero, complement(zero)), meet(complement(zero), top))
% 19.41/3.04 = { by axiom 1 (def_zero) R->L }
% 19.41/3.04 join(zero, meet(complement(zero), top))
% 19.41/3.04 = { by lemma 18 R->L }
% 19.41/3.04 join(zero, meet(top, complement(zero)))
% 19.41/3.04 = { by lemma 24 }
% 19.41/3.04 join(zero, complement(zero))
% 19.41/3.04 = { by axiom 5 (def_top) R->L }
% 19.41/3.04 top
% 19.41/3.04
% 19.41/3.04 Lemma 26: join(X, join(Y, complement(X))) = join(Y, top).
% 19.41/3.04 Proof:
% 19.41/3.04 join(X, join(Y, complement(X)))
% 19.41/3.04 = { by axiom 4 (maddux1_join_commutativity) R->L }
% 19.41/3.04 join(X, join(complement(X), Y))
% 19.41/3.04 = { by axiom 11 (maddux2_join_associativity) }
% 19.41/3.04 join(join(X, complement(X)), Y)
% 19.41/3.04 = { by axiom 5 (def_top) R->L }
% 19.41/3.04 join(top, Y)
% 19.41/3.04 = { by axiom 4 (maddux1_join_commutativity) }
% 19.41/3.04 join(Y, top)
% 19.41/3.04
% 19.41/3.04 Lemma 27: join(top, complement(X)) = top.
% 19.41/3.04 Proof:
% 19.41/3.04 join(top, complement(X))
% 19.41/3.04 = { by axiom 4 (maddux1_join_commutativity) R->L }
% 19.41/3.04 join(complement(X), top)
% 19.41/3.04 = { by lemma 26 R->L }
% 19.41/3.04 join(X, join(complement(X), complement(X)))
% 19.41/3.04 = { by lemma 22 }
% 19.41/3.04 join(X, complement(X))
% 19.41/3.04 = { by axiom 5 (def_top) R->L }
% 19.41/3.04 top
% 19.41/3.04
% 19.41/3.04 Lemma 28: join(zero, meet(X, top)) = X.
% 19.41/3.04 Proof:
% 19.41/3.04 join(zero, meet(X, top))
% 19.41/3.04 = { by lemma 25 R->L }
% 19.41/3.04 join(zero, meet(X, complement(zero)))
% 19.41/3.04 = { by lemma 15 R->L }
% 19.41/3.04 join(complement(top), meet(X, complement(zero)))
% 19.41/3.04 = { by lemma 27 R->L }
% 19.41/3.04 join(complement(join(top, complement(X))), meet(X, complement(zero)))
% 19.41/3.04 = { by lemma 25 R->L }
% 19.41/3.04 join(complement(join(complement(zero), complement(X))), meet(X, complement(zero)))
% 19.41/3.04 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 19.41/3.04 join(meet(zero, X), meet(X, complement(zero)))
% 19.41/3.04 = { by lemma 18 R->L }
% 19.41/3.04 join(meet(X, zero), meet(X, complement(zero)))
% 19.41/3.04 = { by lemma 17 }
% 19.41/3.04 X
% 19.41/3.04
% 19.41/3.04 Lemma 29: join(zero, complement(X)) = complement(X).
% 19.41/3.04 Proof:
% 19.41/3.04 join(zero, complement(X))
% 19.41/3.04 = { by lemma 24 R->L }
% 19.41/3.04 join(zero, meet(top, complement(X)))
% 19.41/3.04 = { by lemma 18 }
% 19.41/3.04 join(zero, meet(complement(X), top))
% 19.41/3.04 = { by lemma 28 }
% 19.41/3.04 complement(X)
% 19.41/3.04
% 19.41/3.04 Lemma 30: meet(X, top) = X.
% 19.41/3.04 Proof:
% 19.41/3.04 meet(X, top)
% 19.41/3.04 = { by lemma 19 R->L }
% 19.41/3.04 complement(join(zero, complement(X)))
% 19.41/3.04 = { by lemma 29 R->L }
% 19.41/3.04 join(zero, complement(join(zero, complement(X))))
% 19.41/3.04 = { by lemma 19 }
% 19.41/3.04 join(zero, meet(X, top))
% 19.41/3.04 = { by lemma 28 }
% 19.41/3.04 X
% 19.41/3.04
% 19.41/3.04 Lemma 31: join(X, X) = X.
% 19.41/3.04 Proof:
% 19.41/3.04 join(X, X)
% 19.41/3.04 = { by lemma 30 R->L }
% 19.41/3.04 join(X, meet(X, top))
% 19.41/3.04 = { by lemma 30 R->L }
% 19.41/3.04 join(meet(X, top), meet(X, top))
% 19.41/3.04 = { by lemma 18 }
% 19.41/3.04 join(meet(top, X), meet(X, top))
% 19.41/3.04 = { by lemma 18 }
% 19.41/3.04 join(meet(top, X), meet(top, X))
% 19.41/3.04 = { by axiom 7 (maddux4_definiton_of_meet) }
% 19.41/3.04 join(meet(top, X), complement(join(complement(top), complement(X))))
% 19.41/3.04 = { by axiom 7 (maddux4_definiton_of_meet) }
% 19.41/3.04 join(complement(join(complement(top), complement(X))), complement(join(complement(top), complement(X))))
% 19.41/3.04 = { by lemma 22 }
% 19.41/3.04 complement(join(complement(top), complement(X)))
% 19.41/3.04 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 19.41/3.04 meet(top, X)
% 19.41/3.04 = { by lemma 18 R->L }
% 19.41/3.04 meet(X, top)
% 19.41/3.04 = { by lemma 30 }
% 19.41/3.04 X
% 19.41/3.04
% 19.41/3.04 Lemma 32: join(X, top) = top.
% 19.41/3.04 Proof:
% 19.41/3.04 join(X, top)
% 19.41/3.04 = { by lemma 27 R->L }
% 19.41/3.04 join(X, join(top, complement(X)))
% 19.41/3.04 = { by lemma 26 }
% 19.41/3.04 join(top, top)
% 19.41/3.04 = { by lemma 26 R->L }
% 19.41/3.04 join(zero, join(top, complement(zero)))
% 19.41/3.04 = { by lemma 27 }
% 19.41/3.04 join(zero, top)
% 19.41/3.04 = { by axiom 4 (maddux1_join_commutativity) R->L }
% 19.41/3.04 join(top, zero)
% 19.41/3.04 = { by lemma 15 R->L }
% 19.41/3.04 join(top, complement(top))
% 19.41/3.04 = { by axiom 5 (def_top) R->L }
% 19.41/3.04 top
% 19.41/3.04
% 19.41/3.04 Lemma 33: join(X, zero) = X.
% 19.41/3.04 Proof:
% 19.41/3.04 join(X, zero)
% 19.41/3.04 = { by lemma 15 R->L }
% 19.41/3.04 join(X, complement(top))
% 19.41/3.04 = { by lemma 32 R->L }
% 19.41/3.04 join(X, complement(join(complement(X), top)))
% 19.41/3.04 = { by lemma 30 R->L }
% 19.41/3.04 join(meet(X, top), complement(join(complement(X), top)))
% 19.41/3.04 = { by lemma 16 }
% 19.41/3.04 X
% 19.41/3.04
% 19.41/3.04 Lemma 34: join(one, x0) = one.
% 19.41/3.04 Proof:
% 19.41/3.04 join(one, x0)
% 19.41/3.04 = { by axiom 4 (maddux1_join_commutativity) R->L }
% 19.41/3.04 join(x0, one)
% 19.41/3.04 = { by axiom 6 (goals) }
% 19.41/3.04 one
% 19.41/3.04
% 19.41/3.04 Lemma 35: join(meet(X, Y), complement(join(Y, complement(X)))) = X.
% 19.41/3.04 Proof:
% 19.41/3.04 join(meet(X, Y), complement(join(Y, complement(X))))
% 19.41/3.04 = { by axiom 4 (maddux1_join_commutativity) R->L }
% 19.41/3.04 join(meet(X, Y), complement(join(complement(X), Y)))
% 19.41/3.04 = { by lemma 16 }
% 19.41/3.04 X
% 19.41/3.04
% 19.41/3.04 Lemma 36: meet(X, join(complement(X), Y)) = meet(X, Y).
% 19.41/3.04 Proof:
% 19.41/3.04 meet(X, join(complement(X), Y))
% 19.41/3.04 = { by axiom 7 (maddux4_definiton_of_meet) }
% 19.41/3.04 complement(join(complement(X), complement(join(complement(X), Y))))
% 19.41/3.04 = { by lemma 16 R->L }
% 19.41/3.04 complement(join(complement(X), complement(join(complement(X), join(meet(Y, complement(X)), complement(join(complement(Y), complement(X))))))))
% 19.41/3.04 = { by axiom 4 (maddux1_join_commutativity) R->L }
% 19.41/3.04 complement(join(complement(X), complement(join(complement(X), join(complement(join(complement(Y), complement(X))), meet(Y, complement(X)))))))
% 19.41/3.04 = { by lemma 22 R->L }
% 19.41/3.04 complement(join(complement(X), complement(join(complement(X), join(join(complement(join(complement(Y), complement(X))), complement(join(complement(Y), complement(X)))), meet(Y, complement(X)))))))
% 19.41/3.04 = { by axiom 11 (maddux2_join_associativity) R->L }
% 19.41/3.04 complement(join(complement(X), complement(join(complement(X), join(complement(join(complement(Y), complement(X))), join(complement(join(complement(Y), complement(X))), meet(Y, complement(X))))))))
% 19.41/3.04 = { by axiom 4 (maddux1_join_commutativity) }
% 19.41/3.04 complement(join(complement(X), complement(join(complement(X), join(complement(join(complement(Y), complement(X))), join(meet(Y, complement(X)), complement(join(complement(Y), complement(X)))))))))
% 19.41/3.04 = { by lemma 16 }
% 19.41/3.04 complement(join(complement(X), complement(join(complement(X), join(complement(join(complement(Y), complement(X))), Y)))))
% 19.41/3.04 = { by axiom 4 (maddux1_join_commutativity) }
% 19.41/3.04 complement(join(complement(X), complement(join(complement(X), join(Y, complement(join(complement(Y), complement(X))))))))
% 19.41/3.04 = { by axiom 4 (maddux1_join_commutativity) }
% 19.41/3.04 complement(join(complement(X), complement(join(complement(X), join(Y, complement(join(complement(X), complement(Y))))))))
% 19.41/3.04 = { by axiom 11 (maddux2_join_associativity) }
% 19.41/3.04 complement(join(complement(X), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 19.41/3.04 = { by lemma 30 R->L }
% 19.41/3.04 complement(join(meet(complement(X), top), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 19.41/3.04 = { by lemma 19 R->L }
% 19.41/3.04 complement(join(complement(join(zero, complement(complement(X)))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 19.41/3.04 = { by lemma 16 R->L }
% 19.41/3.04 complement(join(complement(join(meet(join(zero, complement(complement(X))), Y), complement(join(complement(join(zero, complement(complement(X)))), Y)))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 19.41/3.04 = { by axiom 7 (maddux4_definiton_of_meet) }
% 19.41/3.04 complement(join(complement(join(complement(join(complement(join(zero, complement(complement(X)))), complement(Y))), complement(join(complement(join(zero, complement(complement(X)))), Y)))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 19.41/3.04 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 19.41/3.04 complement(join(meet(join(complement(join(zero, complement(complement(X)))), complement(Y)), join(complement(join(zero, complement(complement(X)))), Y)), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 19.41/3.04 = { by lemma 18 R->L }
% 19.41/3.04 complement(join(meet(join(complement(join(zero, complement(complement(X)))), Y), join(complement(join(zero, complement(complement(X)))), complement(Y))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 19.41/3.04 = { by axiom 4 (maddux1_join_commutativity) }
% 19.41/3.04 complement(join(meet(join(Y, complement(join(zero, complement(complement(X))))), join(complement(join(zero, complement(complement(X)))), complement(Y))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 19.41/3.04 = { by axiom 4 (maddux1_join_commutativity) }
% 19.41/3.04 complement(join(meet(join(Y, complement(join(zero, complement(complement(X))))), join(complement(Y), complement(join(zero, complement(complement(X)))))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 19.41/3.04 = { by lemma 19 }
% 19.41/3.04 complement(join(meet(join(Y, meet(complement(X), top)), join(complement(Y), complement(join(zero, complement(complement(X)))))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 19.41/3.04 = { by lemma 30 }
% 19.41/3.04 complement(join(meet(join(Y, complement(X)), join(complement(Y), complement(join(zero, complement(complement(X)))))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 19.41/3.04 = { by lemma 19 }
% 19.41/3.04 complement(join(meet(join(Y, complement(X)), join(complement(Y), meet(complement(X), top))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 19.41/3.04 = { by lemma 30 }
% 19.41/3.04 complement(join(meet(join(Y, complement(X)), join(complement(Y), complement(X))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 19.41/3.04 = { by axiom 4 (maddux1_join_commutativity) }
% 19.41/3.04 complement(join(meet(join(complement(X), Y), join(complement(Y), complement(X))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 19.41/3.04 = { by axiom 4 (maddux1_join_commutativity) }
% 19.41/3.04 complement(join(meet(join(complement(X), Y), join(complement(X), complement(Y))), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 19.41/3.04 = { by lemma 18 }
% 19.41/3.04 complement(join(meet(join(complement(X), complement(Y)), join(complement(X), Y)), complement(join(join(complement(X), Y), complement(join(complement(X), complement(Y)))))))
% 19.41/3.04 = { by lemma 35 }
% 19.41/3.04 complement(join(complement(X), complement(Y)))
% 19.41/3.04 = { by axiom 7 (maddux4_definiton_of_meet) R->L }
% 19.41/3.04 meet(X, Y)
% 19.41/3.04
% 19.41/3.04 Lemma 37: meet(X, composition(x0, top)) = composition(x0, X).
% 19.41/3.04 Proof:
% 19.41/3.04 meet(X, composition(x0, top))
% 19.41/3.04 = { by lemma 36 R->L }
% 19.41/3.04 meet(X, join(complement(X), composition(x0, top)))
% 19.41/3.04 = { by axiom 5 (def_top) }
% 19.41/3.04 meet(X, join(complement(X), composition(x0, join(complement(X), complement(complement(X))))))
% 19.41/3.04 = { by axiom 2 (converse_idempotence) R->L }
% 19.41/3.05 meet(X, join(complement(X), composition(x0, join(complement(X), converse(converse(complement(complement(X))))))))
% 19.41/3.05 = { by axiom 2 (converse_idempotence) R->L }
% 19.41/3.05 meet(X, join(complement(X), converse(converse(composition(x0, join(complement(X), converse(converse(complement(complement(X))))))))))
% 19.41/3.05 = { by axiom 8 (converse_multiplicativity) }
% 19.41/3.05 meet(X, join(complement(X), converse(composition(converse(join(complement(X), converse(converse(complement(complement(X)))))), converse(x0)))))
% 19.41/3.05 = { by axiom 4 (maddux1_join_commutativity) R->L }
% 19.41/3.05 meet(X, join(complement(X), converse(composition(converse(join(converse(converse(complement(complement(X)))), complement(X))), converse(x0)))))
% 19.41/3.05 = { by axiom 10 (converse_additivity) }
% 19.41/3.05 meet(X, join(complement(X), converse(composition(join(converse(converse(converse(complement(complement(X))))), converse(complement(X))), converse(x0)))))
% 19.41/3.05 = { by axiom 2 (converse_idempotence) }
% 19.41/3.05 meet(X, join(complement(X), converse(composition(join(converse(complement(complement(X))), converse(complement(X))), converse(x0)))))
% 19.41/3.05 = { by axiom 4 (maddux1_join_commutativity) }
% 19.41/3.05 meet(X, join(complement(X), converse(composition(join(converse(complement(X)), converse(complement(complement(X)))), converse(x0)))))
% 19.41/3.05 = { by axiom 12 (composition_distributivity) }
% 19.41/3.05 meet(X, join(complement(X), converse(join(composition(converse(complement(X)), converse(x0)), composition(converse(complement(complement(X))), converse(x0))))))
% 19.41/3.05 = { by axiom 8 (converse_multiplicativity) R->L }
% 19.41/3.05 meet(X, join(complement(X), converse(join(converse(composition(x0, complement(X))), composition(converse(complement(complement(X))), converse(x0))))))
% 19.41/3.05 = { by axiom 4 (maddux1_join_commutativity) R->L }
% 19.41/3.05 meet(X, join(complement(X), converse(join(composition(converse(complement(complement(X))), converse(x0)), converse(composition(x0, complement(X)))))))
% 19.41/3.05 = { by axiom 2 (converse_idempotence) R->L }
% 19.41/3.05 meet(X, join(complement(X), converse(join(composition(converse(converse(converse(complement(complement(X))))), converse(x0)), converse(composition(x0, complement(X)))))))
% 19.41/3.05 = { by axiom 8 (converse_multiplicativity) R->L }
% 19.41/3.05 meet(X, join(complement(X), converse(join(converse(composition(x0, converse(converse(complement(complement(X)))))), converse(composition(x0, complement(X)))))))
% 19.41/3.05 = { by axiom 10 (converse_additivity) R->L }
% 19.41/3.05 meet(X, join(complement(X), converse(converse(join(composition(x0, converse(converse(complement(complement(X))))), composition(x0, complement(X)))))))
% 19.41/3.05 = { by axiom 4 (maddux1_join_commutativity) }
% 19.41/3.05 meet(X, join(complement(X), converse(converse(join(composition(x0, complement(X)), composition(x0, converse(converse(complement(complement(X))))))))))
% 19.41/3.05 = { by axiom 2 (converse_idempotence) }
% 19.41/3.05 meet(X, join(complement(X), join(composition(x0, complement(X)), composition(x0, converse(converse(complement(complement(X))))))))
% 19.41/3.05 = { by axiom 2 (converse_idempotence) }
% 19.41/3.05 meet(X, join(complement(X), join(composition(x0, complement(X)), composition(x0, complement(complement(X))))))
% 19.41/3.05 = { by axiom 11 (maddux2_join_associativity) }
% 19.41/3.05 meet(X, join(join(complement(X), composition(x0, complement(X))), composition(x0, complement(complement(X)))))
% 19.41/3.05 = { by lemma 21 R->L }
% 19.41/3.05 meet(X, join(join(composition(one, complement(X)), composition(x0, complement(X))), composition(x0, complement(complement(X)))))
% 19.41/3.05 = { by axiom 12 (composition_distributivity) R->L }
% 19.41/3.05 meet(X, join(composition(join(one, x0), complement(X)), composition(x0, complement(complement(X)))))
% 19.41/3.05 = { by lemma 34 }
% 19.41/3.05 meet(X, join(composition(one, complement(X)), composition(x0, complement(complement(X)))))
% 19.41/3.05 = { by lemma 21 }
% 19.41/3.05 meet(X, join(complement(X), composition(x0, complement(complement(X)))))
% 19.41/3.05 = { by lemma 36 }
% 19.41/3.05 meet(X, composition(x0, complement(complement(X))))
% 19.41/3.05 = { by lemma 29 R->L }
% 19.41/3.05 meet(X, composition(x0, join(zero, complement(complement(X)))))
% 19.41/3.05 = { by lemma 23 }
% 19.41/3.05 meet(X, composition(x0, X))
% 19.41/3.05 = { by lemma 18 }
% 19.41/3.05 meet(composition(x0, X), X)
% 19.41/3.05 = { by lemma 33 R->L }
% 19.41/3.05 join(meet(composition(x0, X), X), zero)
% 19.41/3.05 = { by lemma 15 R->L }
% 19.41/3.05 join(meet(composition(x0, X), X), complement(top))
% 19.41/3.05 = { by lemma 32 R->L }
% 19.41/3.05 join(meet(composition(x0, X), X), complement(join(composition(meet(one, complement(x0)), X), top)))
% 20.74/3.05 = { by lemma 26 R->L }
% 20.74/3.05 join(meet(composition(x0, X), X), complement(join(composition(x0, X), join(composition(meet(one, complement(x0)), X), complement(composition(x0, X))))))
% 20.74/3.05 = { by axiom 4 (maddux1_join_commutativity) R->L }
% 20.74/3.05 join(meet(composition(x0, X), X), complement(join(composition(x0, X), join(complement(composition(x0, X)), composition(meet(one, complement(x0)), X)))))
% 20.74/3.05 = { by axiom 11 (maddux2_join_associativity) }
% 20.74/3.05 join(meet(composition(x0, X), X), complement(join(join(composition(x0, X), complement(composition(x0, X))), composition(meet(one, complement(x0)), X))))
% 20.74/3.05 = { by axiom 4 (maddux1_join_commutativity) }
% 20.74/3.05 join(meet(composition(x0, X), X), complement(join(composition(meet(one, complement(x0)), X), join(composition(x0, X), complement(composition(x0, X))))))
% 20.74/3.05 = { by axiom 11 (maddux2_join_associativity) }
% 20.74/3.05 join(meet(composition(x0, X), X), complement(join(join(composition(meet(one, complement(x0)), X), composition(x0, X)), complement(composition(x0, X)))))
% 20.74/3.05 = { by axiom 12 (composition_distributivity) R->L }
% 20.74/3.05 join(meet(composition(x0, X), X), complement(join(composition(join(meet(one, complement(x0)), x0), X), complement(composition(x0, X)))))
% 20.74/3.05 = { by axiom 4 (maddux1_join_commutativity) }
% 20.74/3.05 join(meet(composition(x0, X), X), complement(join(complement(composition(x0, X)), composition(join(meet(one, complement(x0)), x0), X))))
% 20.74/3.05 = { by axiom 4 (maddux1_join_commutativity) }
% 20.74/3.05 join(meet(composition(x0, X), X), complement(join(complement(composition(x0, X)), composition(join(x0, meet(one, complement(x0))), X))))
% 20.74/3.05 = { by lemma 35 R->L }
% 20.74/3.05 join(meet(composition(x0, X), X), complement(join(complement(composition(x0, X)), composition(join(join(meet(x0, one), complement(join(one, complement(x0)))), meet(one, complement(x0))), X))))
% 20.74/3.05 = { by lemma 34 R->L }
% 20.74/3.05 join(meet(composition(x0, X), X), complement(join(complement(composition(x0, X)), composition(join(join(meet(x0, one), complement(join(join(one, x0), complement(x0)))), meet(one, complement(x0))), X))))
% 20.74/3.05 = { by axiom 11 (maddux2_join_associativity) R->L }
% 20.74/3.05 join(meet(composition(x0, X), X), complement(join(complement(composition(x0, X)), composition(join(join(meet(x0, one), complement(join(one, join(x0, complement(x0))))), meet(one, complement(x0))), X))))
% 20.74/3.05 = { by axiom 5 (def_top) R->L }
% 20.74/3.05 join(meet(composition(x0, X), X), complement(join(complement(composition(x0, X)), composition(join(join(meet(x0, one), complement(join(one, top))), meet(one, complement(x0))), X))))
% 20.74/3.05 = { by lemma 32 }
% 20.74/3.05 join(meet(composition(x0, X), X), complement(join(complement(composition(x0, X)), composition(join(join(meet(x0, one), complement(top)), meet(one, complement(x0))), X))))
% 20.74/3.05 = { by lemma 15 }
% 20.74/3.05 join(meet(composition(x0, X), X), complement(join(complement(composition(x0, X)), composition(join(join(meet(x0, one), zero), meet(one, complement(x0))), X))))
% 20.74/3.05 = { by lemma 33 }
% 20.74/3.05 join(meet(composition(x0, X), X), complement(join(complement(composition(x0, X)), composition(join(meet(x0, one), meet(one, complement(x0))), X))))
% 20.74/3.05 = { by lemma 18 R->L }
% 20.74/3.05 join(meet(composition(x0, X), X), complement(join(complement(composition(x0, X)), composition(join(meet(one, x0), meet(one, complement(x0))), X))))
% 20.74/3.05 = { by lemma 17 }
% 20.74/3.05 join(meet(composition(x0, X), X), complement(join(complement(composition(x0, X)), composition(one, X))))
% 20.74/3.05 = { by lemma 21 }
% 20.74/3.05 join(meet(composition(x0, X), X), complement(join(complement(composition(x0, X)), X)))
% 20.74/3.05 = { by axiom 4 (maddux1_join_commutativity) }
% 20.74/3.05 join(meet(composition(x0, X), X), complement(join(X, complement(composition(x0, X)))))
% 20.74/3.05 = { by lemma 35 }
% 20.74/3.05 composition(x0, X)
% 20.74/3.05
% 20.74/3.05 Goal 1 (goals_1): tuple(join(meet(composition(x0, top), x1), composition(x0, x1)), join(composition(x0, x1), meet(composition(x0, top), x1))) = tuple(composition(x0, x1), meet(composition(x0, top), x1)).
% 20.74/3.05 Proof:
% 20.74/3.05 tuple(join(meet(composition(x0, top), x1), composition(x0, x1)), join(composition(x0, x1), meet(composition(x0, top), x1)))
% 20.74/3.05 = { by axiom 4 (maddux1_join_commutativity) }
% 20.74/3.05 tuple(join(composition(x0, x1), meet(composition(x0, top), x1)), join(composition(x0, x1), meet(composition(x0, top), x1)))
% 20.74/3.05 = { by lemma 18 R->L }
% 20.74/3.05 tuple(join(composition(x0, x1), meet(x1, composition(x0, top))), join(composition(x0, x1), meet(composition(x0, top), x1)))
% 20.74/3.05 = { by lemma 18 R->L }
% 20.74/3.05 tuple(join(composition(x0, x1), meet(x1, composition(x0, top))), join(composition(x0, x1), meet(x1, composition(x0, top))))
% 20.74/3.05 = { by lemma 37 }
% 20.74/3.05 tuple(join(composition(x0, x1), composition(x0, x1)), join(composition(x0, x1), meet(x1, composition(x0, top))))
% 20.74/3.05 = { by lemma 37 }
% 20.74/3.05 tuple(join(composition(x0, x1), composition(x0, x1)), join(composition(x0, x1), composition(x0, x1)))
% 20.74/3.05 = { by lemma 31 }
% 20.74/3.05 tuple(composition(x0, x1), join(composition(x0, x1), composition(x0, x1)))
% 20.74/3.05 = { by lemma 31 }
% 20.74/3.05 tuple(composition(x0, x1), composition(x0, x1))
% 20.74/3.05 = { by lemma 37 R->L }
% 20.74/3.05 tuple(composition(x0, x1), meet(x1, composition(x0, top)))
% 20.74/3.05 = { by lemma 18 }
% 20.74/3.05 tuple(composition(x0, x1), meet(composition(x0, top), x1))
% 20.74/3.05 % SZS output end Proof
% 20.74/3.05
% 20.74/3.05 RESULT: Theorem (the conjecture is true).
%------------------------------------------------------------------------------